Start with , an intersection of two maximal subgroups of . Suppose and are maximal subgroups of such that is maximal among all intersections of two maximal subgroups that contain , i.e., it is not properly contained in any other intersection of two maximal subgroups of . Such a maximal intersection exists by the finiteness of . Consider the subgroup . This contains strictly (by Fact (1) and the fact that is nilpotent on account of being proper in ) and is contained in . However, it is not contained in any maximal subgroup of other than by the maximality assumption on . Similarly, is not contained in any maximal subgroup of other than . Thus, , which contains both and , is not contained in any maximal subgroup of . Because of the finiteness of , it must be the whole group . Thus, is normal in .

By Step (1), the intersection is contained in an intersection of two maximal subgroups wherein the latter intersection is normal. Moreover, the latter intersection is also a proper subgroup. Since is simple, the latter intersection must be the trivial subgroup, and therefore the original intersection must also be the trivial subgroup.